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1
Content available remote Nowe wyzwania dla polskiej kryptologii drugiej dekady XXI wieku
Bondaryk K., Pomykała J.
Studia Bezpieczeństwa Narodowego
|
2014
|
R. 4, Nr 6
19--31
PL
W artykule analizujemy wyzwania dla polskiej kryptologii XXI wieku ze szczególnym uwzględnieniem potrzeb narodowej kryptologii i roli, jaką spełniają w niej wybrane dziedziny matematyki, takie jak teoria liczb i geometria algebraiczna. W szczególności pokreślono rolę i bezpieczeństwo kryptosystemów bazujących na iloczynach dwuliniowych, a także problemy złożoności obliczeniowej ważnych dla kryptologii algorytmów deterministycznych. Wskazano na znaczenie funkcji typu L we współczesnej kryptografii i kryptoanalizie.
EN
In this paper we analyze the challenges for the twenty-first century Polish cryptology with special emphasis on the needs of the national cryptology and the role they perform in the selected areas of mathematics such as number theory and algebraic geometry. In particular, westress the role and security of bilinear based cryptosystems, as well as the problems of computational complexity of deterministic algorithms important for cryptology. We pointed out the importance of L-functions in modern cryptography and cryptoanalysis.
2
Content available remote Geometria rozmaitości modułów
Bobiński G.
Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne
|
2006
|
[Z] 42
3-13
3
Content available remote Życie i dzieło Alexandra Grothendiecka
Pragacz P.
Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne
|
2004
|
[Z] 40
107-127
4
Content available remote Program Langlandsa według Lafforgue'a
Langer A.
Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne
|
2003
|
[Z] 39
39-46
5
Content available remote On the moment measure conjecture
Białynicki-Birula A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2002
|
Vol. 50, no 1
33-40
EN
The main purpose of the paper is to give a characterization of such open invariant subsets U of projective spaces with an action of a torus T, which admit a good quotient U --> U//T and the quotient space is compact. It follows from the result that moment measure conjecture (stated in [4]) is valid in the case of projective spaces (and their products).
6
Content available remote A complement of a hypersurface in affine variety
Jelonek Z.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2001
|
Vol. 49, no 4
375-379
EN
We construct general examples of affine normal varieties X which have a hypersurface H [is a subset of] X, such that the variety X [...] H is not affine. We also show, that if X is an affine normal surface and H is a curve in X, then X [...] H is affine, too.
7
Content available remote The Whitney property of a fiber of a definable mapping
Kocel-Cynk B.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2000
|
Vol. 48, no 2
141-147
EN
We prove the following theorem Let S be a polynomially bounded o-minimal structure on (R,+,.) and let f : A --> [R^n] be a continuous, definable function on a compact definable set [A is subset of R^m]. Then there exist a positive real number [alpha belongs to R+] and a definable function C : f(A) --> R+ such that for any x [belongs to] f(A) and any two points p and q in the same connected component of [f^-1](x) there exists a piece-wise analytic curve gamma joinning p and q in [f^-1](x) with length [gamma is less than or equal to C(x)][...]. As a consequence we obtain the regular separation with parameter for definable sets.
8
Content available remote Product formulas for the Milnor class
Ohmoto T., Yokura S.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2000
|
Vol. 48, no 4
387-401
EN
Vie give a general product formula for the Milnor class. Using this formula we obtain a Parusiński-Pragacz-type formula for the Milnor class of a finite Cartesian product of hypersurfaces and we show that in general it is necessary to also consider the Chern classes of subbundles of the total vector bundle to obtain a Parusiński-Pragacz-type formula for the Milnor class of general local complete intersections. Furthermore, we also give another kind of product formula for the Milnor class of singular hypersurfaces by using the so-called Thom-Sebastiani construction.
9
Content available remote Injections into affine hypersurfaces
Jelonek Z.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2000
|
Vol. 48, no 1
63-67
EN
Let X be a smooth affine variety of dimension n > 2. Assume that the group H[sub 1](X,Z) is a torsion group and that [chi](X) = 1. Let Y be a projectively smooth affine hypersurface Y [is a subset of] C[sup n+1] of degree d > 1, which is smooth at infinity. Then there is no injective polynomial mapping f : X --> Y. This contradicts a result of Peretz [5].
10
Content available remote A geometry of polynomial transformations of the real plane
Jelonek Z.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2000
|
Vol. 48, no 1
57-62
EN
In this paper we study the set of points at which a real polynomial mapping of the plane is not proper.
11
Content available remote A geometry of Pinchuk's map
Goździewicz G.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2000
|
Vol. 48, no 1
69-75
EN
Sergey Pinchuk found a polynomial map from the real plane to itself which is a local diffeomorphism but is not one-to-one. The aim of this paper is to give a geometric description of Pinchuk's map.
12
Content available remote Smooth affine curves without embeddings into the affine plane
Jelonek Z.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1999
|
Vol. 47, no 4
363-367
EN
In this paper we gives examples of smooth affine curves which cannot be embedded in [C^2].
13
Content available remote A number of points in the set [C^2 difference of sets F(C^2)]
Jelonek Z.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1999
|
Vol. 47, no 3
257-261
EN
In this paper we estimate a number of points of the set [C^2 difference of sets F(C^2)] for a polynomial quasi-finite mapping F : [C^2 --> C^2].
14
Content available remote On the birational structure of exceptional irreducible components of moduli schemes of rank 2 vector bundles on algebraic surfaces with negative Kodaira dimension
Ballico E.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
|
Vol. 46, no 1
17-23
EN
Here we study the birational structure of "exceptional" irreducible components of moduli schemes of rank 2 vector bundles on algebraic surfaces with negative Kodaira dimension. In particular for every rational surface X prove the existence of several such components which are rational.
15
Content available remote On semialgebraic tangent cones
Fortuna E.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
|
Vol. 46, no 4
383-390
EN
The paper deals with the following question: which semialgebraic subsets of [R^n] can be realized as tangent cones to real algebraic subsets of [R^n]? At first we prove that the answer is positive for every closed semialgebraic cone in [R^n] of dimension [is less than or equal to 2]. Then, for closed semialgebraic cones A in [R^n] admitting a sufficiently nice algebraic presentation, we explicitely give equations of an algebraic subset V [...] [R^n] having A as its tangent come.
16
Content available remote Linearly independent divisors on complex manifolds
Jelonek Z.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
|
Vol. 46, no 1
25-30
EN
Let X be a connected normal complex space and let D be a non-zero Cartier divisor on X with the support [...]. We show that if D is a principal divisor then the group H[sub 1](X \ [absolute value of D],Z) cannot be a torsion group. In particular the group H[sub1](H \ [absolute value of D],Z)] must be infinite. As a corollary we prove that simple Cartier divisors D[sub 1],...,D[sub r] on a complex manifold X are linearly independent in Cl(X), provided the group H[sub1](X \ [...] is a torsion group.
17
Content available remote Effective Nullstellensatz on analytic and algebraic varieties
Płoski A., Tworzewski P.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
|
Vol. 46, no 1
31-38
EN
Sharp estimates of the Nullstellensatz exponent on analytic and algebraic sets are given.
18
Content available remote A model-theoretic criterion for quantifier elimination and its application to geometry
Nowak K.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
|
Vol. 46, no 4
377-381
EN
In this paper we present a model-theoretic criterion for quantifier elimination being a variant of Shoenfield's theorem (see [1], Chap. V, [paragraph]5). Our short proof is based directly on Godel's completeness and compactness theorems as well as on the concept of diagrams, and does not involve model-completeness or Robinson's test as does for instance the proof of certain related criteria given in [2], Chap. VIII, [paragraph]4. As a consequence, we immediately obtain the theorems of Chevalley and Tarski-Seidenberg from algebraic and semialalgebraic geometry.
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